The influence of seismic frequency spectrum on the instability of loess slope

The input of seismic wave with different frequency has a significant impact on loess slope instability. On the basis of field investigation and experiments, the particle flow software PFC2D was used to explore the effect of seismic frequency spectrum on slope instability through the process of calibrating soil microscopic parameters, model establishment, seismic wave input and other processes. The results show that: 1. The low-frequency component of the input wave is the main frequency band that causes the slope instability, the slope has amplifying effect on the low-frequency input wave, and the slope has a "filtering" effect on the high-frequency input wave; 2. The instability of the slope will cause an increase in frequency components above 10 Hz; 3. The special structure of the slope is one of the main reasons for the instability of the slope. This result has theoretical and practical significance for earthquake landslide prevention and monitoring and early warning.

The occurrence of seismic landslide is essentially the action of seismic waves on the slope, which causes the slope to change from a static state to a moving state 1 . The violent vibration causes the rock and soil structure to loosen and deform, and slip plane is penetrated 2,3 . Natural seismic waves contain many frequency bands 4 . Although seismic waves of any frequency will cause the slope to vibrate, when the frequency of the earthquake is close to or the same as the natural frequency of the slope, the seismic waves will be amplified several times or even dozens of times due to resonance, and the stress and displacement deformation of the slope will increase, the probability of a landslide will increase accordingly 5,6 .
Many practical investigations and theoretical studies of earthquake landslides show that the three elements (amplitude, frequency spectrum and duration) of seismic waves determines the degree of slope damage in the earthquake 7,8 .Since seismic frequency spectrum is complex, the study of seismic frequency spectrum on slope dynamic response is not sufficient, but facts and existing studies have shown that seismic waves with different frequency spectrum has significant influence on slope stability. Liu et al. 9 confirmed that the seismic frequency spectrum have a certain influence on the slope stability by using the finite method; Xu et al. 10 showed that the slope soil has an amplifying effect on the low-frequency part of the input seismic waves and a filtering effect on the high frequency part by useing FLAC3D. Zhang et al. 11 analyzed the acceleration and displacement response of the slope model under different frequency spectrum seismic waves, and showed that when the input seismic wave characteristic period is greater than 0.65 s, the slope displacement response increases significantly, and the PGA amplification factor of the slope shows an increasing trend. Shi et al. 12 uses wavelet packet analysis to decompose the dynamic earth pressure time-domain curve based on large-scale shaking table experiments, confirming that low-frequency waves play a dominant role in the dynamic earth pressure response, and the slope has a filtering effect on the high frequency part of the seismic wave. Chang et al. 13 simulated the failure process of loess seismic landslide based on PFC, and confirmed that low-frequency vibration had a stronger destructive effect on landslide, and the high-frequency component of slope vibration increased after failure.
The occurrence of landslide is a process of rock and soil damage and movement with sliding, translation and rotation. Finite element, finite difference and other continuum mechanics methods have limitations. On the basis of field investigation and experiments, the particle flow software PFC2D was used to explore the effect of ground motion spectrum on slope instability through the process of calibrating soil microscopic parameters, model establishment, power input and other processes. The main frequency bands that cause slope instability are obtained by analysis.  14,15 . Due to the special geographical location and climatic conditions, most of these loess earthquake landslides are still well-preserved 16

Calculation parameters.
The direct shear test model is used to calibrate the meso-mechanical parameters.
The size of the model is 500 m × 400 m (Fig. 4), and the total number of particles generated is 1707. The particle-to-particle contact model selects the parallel bond (linearpond) model, and the particle meso-mechanical parameters of loess and mudstone are listed in Table 1. Taking loess as an example, under the conditions of meso-mechanical parameters in Table 1, vertical stresses of 100, 200, 300, and 400 kPa were applied to the direct shear model to obtain the shear stress-displacement relationship curve (Fig. 5). Then draw the direct shear test strength curve (Fig. 6), and calculate the macro-mechanical parameters of the material in the direct shear test. The macro-mechanical parameters of rock and soil obtained by numerical simulation of direct shear test. Cohesion c = 26 kPa and internal friction angle φ = 15°. By comparing with c = 21.3 ~ 28.7 kPa and φ = 11.2 ~ 18.3° obtained by indoor physical test, it can be seen that the mechanical parameters obtained by numerical simulation are within its range, and the meso-parameters are available.     Fig. 7. The wall-ball method is used in PFC 2D to establish the original landslide model. The specific steps are as follows: ①Create a rectangular area larger than the column profile in PFC2D and generate particles; ②Import the landslide profile established in AutoCAD, and group the particles according to the landslide boundary ③According to the meso-mechanical parameters obtained by parameter calibration, the particles are assigned; ④Gravity is applied to make the landslide reach the initial stress equilibrium state under the action of gravity; ⑤The particles outside the section area are deleted; ⑥Steps 4 and 5 are repeated until the particles are complete Reconcile with the profile view to achieve a state of stress balance. According to the above steps, the PFC2D model of the Subao landslide is obtained, as shown in Fig. 8. The model is 1500 m long, 230 m high, and has 8998 particles. Under the action of gravity, the slope is stable.

Seismic waves input
In order to explore the influence of the seismic frequency spectrum on the instability of the loess slope, the Kobe seismic wave (NS direction) recorded by the Hanshin earthquake is selected as the original seismic data. The peak acceleration of the Kobe wave is 0.3347 g. Time history, displacement time history and the Fourier spectrum are shown in the Fig. 9. It can be clearly seen from the figure that the main frequency of the kobe wave    Table 2).

Results
In order to monitor the acceleration, velocity, displacement and spectrum change characteristics of the slope under the dynamic action of different frequency spectrums, the movement of particles of different heights at 10 m, 50 m, 90 m, 130 m and 170 m within the slope were monitored. The location is shown in Fig. 11.  In order to further illustrate the influence of the seismic frequency spectrum on the peak acceleration, the peak acceleration magnification under different frequency seismic waves input is plotted in the Fig. 13. It can be clearly seen from the figure that the magnifications of monitoring point 1 are all close to 1, which indicated the dynamic boundary conditions are reasonable. Under the input with the excellent frequency of 0-2 Hz, the amplification of monitoring point 5 is the largest, which can reach 17.9. Under the input with the excellent frequency of 2-4 Hz, 4-6 Hz, 6-8 Hz and 8-10 Hz, The amplifications of the monitoring point 5 are all less than 1, maintaining a stable state. The magnification of monitoring point 4 is larger than that of other points, which is related to its location where is the interface.

Spectrum.
In order to reveal the influence of the frequency spectrum on the loess slope, the Fourier spectrum of the acceleration at the 5 detection points under 5 inputs is calculated and shown in the Fig. 14. It can be seen from the figure that the Fourier spectra of the five detection points have all major changes from bottom to surface no matter what frequency the input is, showing the unique spectral characteristics of the slope itself, the closer to the slope. The closer to the slope surface, the more high-frequency components. The smaller the input frequency, the larger the Fourier amplitude spectrum of the high-frequency components of the slope, which indicates that the collision frequency of particles in the slope increases, the acceleration changes more drastically, and the instability failure occurs. The frequency of the input power is different, and the amplification of the same frequency range at different locations is different. Under the input of the excellent frequency of 0-2 Hz,  Figure 11. The measure locations.   www.nature.com/scientificreports/ 1 and 2 are close to zero. Under the power inputs with excellent frequencies of 2-4 Hz, 4-6 Hz, 6-8 Hz and 8-10 Hz, the trend of the velocity curves of monitoring points 1-4 is consistent, and the velocity of monitoring point 5 appears more "positive" than "negatie", and these points and zero axis form a larger area, which is the performance of the displacement caused by the instability of the slope particles. Under the input with excellent frequencies of 4-6 Hz, 6-8 Hz and 8-10 Hz, the speed amplitude of monitoring point 5 is the smallest value among the 5 monitoring points, indicating that under the action of soil damping, much energy is consumed, which lead the destructive power weak. In order to further compare the displacement of the slope under the 5 kinds of input, the final displacement of the slope is shown in the Fig. 17. It can be clearly seen from the figure that under the input with an excellent frequency of 0-2 Hz, the slope is obviously unstable. The upper loess and the lower mudstone have an obvious displacement difference, and the sliding surface mainly relies on the stratum interface, and the overall shape is irregular arc. As the frequency of input power increases, the degree of slope instability becomes smaller. Under the input of excellent frequency of 2-4 Hz, there is only instability in part of the slope surface, and it is concentrated at the toe and shoulder, where are usually the concentrated areas of tensile failure and compression failure. Under the input with excellent frequencies of 4-6 Hz, 6-8 Hz and 8-10 Hz, only a small range of instability occurs at the toe of the slope, which has no obvious impact on the slope as a whole.

Conclusion
In order to explore the impact of seismic frequency spectrum on slope instability, on the basis of field investigation and experiments, the particle flow software PFC 2D was used to explore the effect of ground motion spectrum on slope instability through the process of calibrating soil microscopic parameters, model establishment, power input and other processes, and obtained the following conclusions through analysis: 1. The frequency spectrum characteristics of seismic wave have a significant impact on the instability of loess slopes. The low-frequency components (0-2 Hz) in the input seismic wave are the main frequency bands that cause slope instability and destruction. Under low-frequency input, the peak acceleration, acceleration amplification factor, speed and displacement near the slope surface are relatively large, and obvious instability and damage phenomena occur; under high-frequency input, the peak acceleration, acceleration amplification factor, speed and displacement near the slope surface are all small, and the slope has a "filtering" effect on the input wave. 2. The destruction of the slope will cause an increase in frequency components above 10 Hz. Regardless of the input seismic wave in any frequency band, under the dual influences of the earthquake and the empty surface, the collision between the broken soil and the original slope is intensified, which further causes the instability and damage of the slope. Due to the "filtering" effect of the slope on the high-frequency components of seismic wave and the increase in high-frequency components caused by rupture, the high-frequency components of seismic wave recorded on slopes mostly reflect the damage of the slope. This conclusion can be used for monitoring and early warning of earthquake landslides. 3. The special structure of the slope is one of the main reasons for the instability of the slope. Due to the sudden change of soil properties and wave impedance, the acceleration and velocity increase at the contact surface of mudstone and loess, which is easy to lose stability at the junction and form a sliding surface. Regardless of the frequency of seismic wave input, the toe of the slope is the first part to lose stability. In the prevention of earthquake landslide disasters, the reinforcement treatment of slope contact surface and slope toe requires attention.
It is worth pointing out that the particle flow method in this paper does not consider the structural properties of soil and the effects of groundwater in the simulation. According to the experience of field investigations and laboratory tests, the structural properties and water sensitivity of loess play an important role in slope instability, which should be taken into consideration in the numerical simulation. The conclusion of this article is the regularity of seismic frequency spectrum on the instability of loess slopes. In future research, the structural and water sensitivity of the soil should be fully considered to quantify the impact of ground motion frequency spectrum on the instability of slopes.

Data availability
The datasets used and/or analysed during the current study available from the corresponding author on reasonable request.